Proper affine deformations of positive representations

TL;DR

This paper introduces a new family of proper affine actions derived from positive Anosov representations into SO(2n,2n-1), constructing fundamental domains and showing the resulting manifolds are handlebodies.

Contribution

It defines a novel class of affine deformations for positive representations and constructs explicit fundamental domains bounded by generalized crooked planes.

Findings

Proper affine actions are constructed for positive Anosov representations.

Fundamental domains are explicitly built using generalized crooked planes.

Resulting quotient manifolds are homeomorphic to handlebodies.

Abstract

We define for every positive Anosov representation of a nonabelian free group into $\mathrm{SO}(2n,2n-1)$ a family of $\mathbb{R}^{4n-1}$-valued cocycles which induce proper affine actions on $\mathbb{R}^{4n-1}$. We construct fundamental domains in $\mathbb{R}^{4n-1}$ bounded by generalized crooked planes for these affine actions, and deduce that the quotient manifolds are homeomorphic to handlebodies.